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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Mass of the body, $m = 1\,\text{kg}$
• Initial velocity, $u = 20\,\text{m/s}$
• Acceleration due to gravity, $g = 10\,\text{m/s}^2$
• Height actually attained, $h' = 18\,\text{m}$
Step 2: Calculate the Initial Kinetic Energy
The initial kinetic energy $K_{i}$ of the body is given by
$$K_{i} = \frac{1}{2} m u^2.$$
Substituting the given values:
$$K_{i} = \frac{1}{2} \times 1 \times (20)^2 = 200\,\text{J}.$$
Step 3: Compute the Ideal Maximum Height (Without Friction)
If there were no energy loss, all of this kinetic energy would convert into potential energy at the maximum height. The potential energy at the maximum height, $mgh$, equals the initial kinetic energy:
$$mgh = 200 \,\text{J}.$$
Since $m = 1\,\text{kg}$ and $g = 10\,\text{m/s}^2$, we have:
$$1 \times 10 \times h = 200 \,\text{J} \quad \Rightarrow \quad h = 20\,\text{m}.$$
Thus, ideally, the body should reach 20 m if no energy were lost.
Step 4: Calculate the Actual Potential Energy
The body actually reaches a height of $h' = 18\,\text{m}$. Therefore, the potential energy at this real peak is:
$$m g h' = 1 \times 10 \times 18 = 180\,\text{J}.$$
Step 5: Determine the Energy Lost due to Air Friction
The energy lost to air friction is the difference between the initial kinetic energy and the potential energy attained at 18 m:
$$\text{Energy lost} = K_{i} - (m g h').$$
Substituting the values:
$$\text{Energy lost} = 200 - 180 = 20\,\text{J}.$$
Step 6: Conclusion
The energy lost due to air friction is $20\,\text{J}$.
